We have seen that the topology of quasicrystals originates from a competition between length scales. Following the approximant path, we understood that this competition “locks” the spectrum of the system into a very specific arrangement with well-defined integrated density of states, as well as associated bulk responses. The latter are quantized and serve for a topological classification of quasicrystals with corresponding bulk and boundary observable effects. In this sense, quasiperiodic order can be thought of as a very specific crystalline symmetry that allows for an enriched topological classification, akin to other crystalline symmetries [1]}. At the same time, the topological characterization discussed here has only been possible for certain types of quasicrystals and under specific assumptions. There is, therefore, still much to explore in this rich zoo of quasiperiodic models. Furthermore, with the advent of controllable shaping of various metamaterials for optical, acoustic, and matter waves the topological properties of quasicrystals are now observable, allowing for new discoveries and applications.
